On Singular Stationarity II (tight stationarity and extenders-based methods)

We study the notion of tightly stationary sets which was introduced by Foreman and Magidor in \cite{ForMag-MS}. We obtain two consistency results which show that it is possible for a sequence of regular cardinals $( \kappa_n )_{n < \omega}$ to have the property that for every sequence $\vec{S}$, of some fixed-cofinality stationary sets $S_n \subseteq \kappa_n$, $\vec{S}$ is tightly stationary in a generic extension. The results are obtained using variations of the short-extenders forcing method.